False positives and negatives

when they both say true the probability is 60*90/100=54;

when the both say false at same time is 40*10/100=4;

so the say always the same fact is 54+4=58%;

so the percentage of cases that they contradict each other is 100-58=42%

The two cases when they contradict each other is:-

(i) A telling the truth and B telling a lie.

(ii) A telling a lie and B telling the truth.

for (i) (60% * 10%) =0.06

for (ii) (40% * 90%) =0.36

either (i) happnes OR (ii) happens

so we get (i) OR (ii)

=> (i) + (ii)

=> 0.06 + 0.36

= 0.42

and in percentage it is 42%

I disagree with all solutions because when they say false at the same time there is a possibility that they say different lie , in that case too they may contradict each other. For instance in the random experiment of throwing a die if the outcome is say 3, then the persons may contradict each other by saying a lie too, say for instance A reports the outcome to be 4 while B reports it to be 6, then they both are telling a lie as well as contradicting each other, you have missed those cases.

Notice that one has to be telling the truth and the other one has to be lying. Knowing that we can get the probability of A intersection Not B and B intersection Not A by multiplying them and the andding the results. P(A) P(NotB) + P(B) P(NotA) Plugging in the numbers we get that: (.9)(1-.6)+(.6)(1-.9) The probability is .42, or 42%.

Python:

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contradictions = 0.0
statements = 0.0
for a in xrange(1, 11):
    statement_a = a <= 6
    for b in xrange(1, 11):
        statement_b = b <= 9
        if statement_a != statement_b:
            contradictions += 1
        statements += 1
print "Answer:", 100*contradictions/statements

Bayes theorem P(a)/P(a)+P(b) 0.6/0.6+0.9 =0.42

I am confused about your question. It takes a chance that both of them are lying but in different way. Then we can't provide the exactly answer.

Notice that either (person A is telling the truth and person B is lying) or (person B is telling the truth and person A is lying). This is the same as (90% (100-60)%)+(60% (100-90)%)=36%+6%=42% I will add latex later.